Expected Value of a Discrete Random Variable with Power Function PMF

In this article, we will delve into the concept of expected value for a discrete random variable whose probability mass function (PMF) is given by the power function. We will explore the methodology to calculate the expected value and discuss the conditions under which the PMF is valid.

Introduction to Discrete Random Variables and PMF

A discrete random variable (X) takes on specific, distinct values. The PMF, denoted as (P_X(x)), gives the probability that (X) takes a particular value (x). For a valid PMF, the sum of the probabilities over all possible values of (X) must equal 1.

The Given PMF and Its Validity

Let us consider a discrete random variable (X) with the PMF defined as:

[P_X(x) frac{1}{x^n}]

Here, (n) is a positive integer and (x) takes values from 1 to infinity. For the PMF to be valid, the sum of all probabilities must equal 1:

[sum_{x1}^{infty} P_X(x) sum_{x1}^{infty} frac{1}{x^n}]

This sum is known as the Zeta function, denoted as (zeta(n)), which is defined for (text{Re}(n) > 1). Thus, the PMF (P_X(x) frac{1}{x^n}) is valid for (text{Re}(n) > 1).

Expected Value Calculation

The expected value (E(X)) of a discrete random variable is calculated using the formula:

[E(X) sum_{x1}^{infty} x cdot P_X(x)]

Substituting the PMF into the formula, we get:

[E(X) sum_{x1}^{infty} x cdot frac{1}{x^n} sum_{x1}^{infty} frac{1}{x^{n-1}}]

To solve this sum, we use the fact that:

[sum_{x1}^{infty} frac{1}{x^{n-1}} zeta(n-1)]

Therefore, the expected value (E(X)) can be expressed as:

[E(X) zeta(n-1)]

Special Cases and Generalization

For specific values of (n), the Zeta function can be evaluated with closed forms. For instance:

(n 2): (zeta(1)) is divergent, but (zeta(2) frac{pi^2}{6}) (n 3): (zeta(2) frac{pi^2}{6}) (n 4): (zeta(3)) is a known but not expressible in terms of elementary functions

In general, the expected value for a discrete random variable with a PMF defined by (P_X(x) frac{1}{x^n}) is:

[E(X) zeta(n-1)]

Provided that (text{Re}(n) > 1).

Conclusion

The expected value of a discrete random variable with the given power function PMF is a function of the Zeta function. Understanding this relationship is crucial for various applications in probability theory and statistics. The conditions under which the PMF is valid, and the evaluation of the expected value for specific values of (n), provide valuable insights into the behavior of such random variables.